In Exercises 1–10, use substitution to determine whether the given x-value is a solution of the equation. __ √ 3 5𝝅tan 2x = ﹣--------- , x = --------- 3 12

Verified step by step guidance

1. Step 1: Start by substituting the given value of \( x = \frac{5\pi}{12} \) into the equation \( \tan(2x) = -\frac{\sqrt{3}}{3} \).
2. Step 2: Calculate \( 2x \) by multiplying \( x \) by 2: \( 2x = 2 \times \frac{5\pi}{12} = \frac{10\pi}{12} = \frac{5\pi}{6} \).
3. Step 3: Evaluate \( \tan\left(\frac{5\pi}{6}\right) \). Recall that \( \tan\left(\frac{5\pi}{6}\right) \) is equivalent to \( \tan\left(\pi - \frac{\pi}{6}\right) \), which is \( -\tan\left(\frac{\pi}{6}\right) \).
4. Step 4: Use the known value \( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \), so \( \tan\left(\frac{5\pi}{6}\right) = -\frac{1}{\sqrt{3}} \).
5. Step 5: Simplify \( -\frac{1}{\sqrt{3}} \) to \( -\frac{\sqrt{3}}{3} \) and compare it to the right side of the equation. Since they match, \( x = \frac{5\pi}{12} \) is a solution.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. The tangent function, specifically, is defined as the ratio of the opposite side to the adjacent side. Understanding how to evaluate these functions at specific angles is crucial for solving trigonometric equations.

Substitution Method

The substitution method involves replacing a variable in an equation with a specific value to determine if the equation holds true. In this context, substituting the given x-value into the equation allows us to check if both sides of the equation are equal, thereby verifying if it is a solution.

Solving Trigonometric Equations

Solving trigonometric equations requires manipulating the equation to isolate the variable, often using identities and algebraic techniques. This process may involve simplifying expressions, applying inverse functions, or using known values of trigonometric functions to find solutions that satisfy the equation.